Question

F(x)=f(x)(1+|sinx|),F(x),f(x)在x=0处可导,求f(0)

Answer 1

F'(0)=lim{x->0}[f(x)(1+|sinx|)-f(0)]/x=lim{x->0}[f(x)-f(0)](1+|sinx|)/x+lim{x->0}[f(0)(1+|sinx|)-f(0)]/x=f'(0)+lim{x->0}f(0)|sinx|/x

左极限lim{x->0-}f(0)|sinx|/x=-f(0),右极限lim{x->0+}f(0)|sinx|/x=f(0).

因为F'(0)存在,所以左导数等于右导数,从而f'(0)-f(0)=f'(0)+f(0),因此f(0)=0.